Aspect ratio dependence of charge transport in turbulent electroconvection

We present measurements of the normalized charge transport or Nusselt number $\rm Nu$ as a function of the aspect ratio $\Gamma$ for turbulent convection in an electrically driven film. In analogy with turbulent Rayleigh-B{\'e}nard convection, we develop the relevant theoretical framework in which we discuss the local-power-law-scaling of $\rm Nu$ with a dimensionless electrical forcing parameter $\cal R$. For these experiments where $10^4 \lesssim {\cal R} \lesssim 2 \times 10^5$ we find that ${\rm Nu} \sim {\rm F}(\Gamma){\cal R}^\gamma$ with $\gamma$ either $ = 0.26 (\pm 0.02)$ or $0.20 (\pm 0.03)$, in excellent agreement with the theoretical predictions of $\gamma = 1/4$ and 1/5. Our measurements of the aspect ratio-dependence of ${\rm Nu}$ for $0.3 \leq \Gamma \leq 17$ compares favorably with the function ${\rm F}(\Gamma)$ from the scaling theory.